reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem Th25:
  [.a * b,c.] = [.a,c.] |^ b * [.b,c.]
proof
  thus [.a * b,c.] = ((a * b)" * c") * (a * b * c) by Th16
    .= (b" * a" * c") * (a * b * c) by GROUP_1:17
    .= (b" * a" * c") * (a * 1_G * b * c) by GROUP_1:def 4
    .= (b" * a" * c") * (a * (c * c") * b * c) by GROUP_1:def 5
    .= (b" * a" * c") * (a * (c * 1_G * c") * b * c) by GROUP_1:def 4
    .= (b" * a" * c") * (a * (c * (b * b") * c") * b * c) by GROUP_1:def 5
    .= b" * (a" * c") * (a * (c * (b * b") * c") * b * c) by GROUP_1:def 3
    .= b" * (a" * c") * (a * (c * (b * b") * c") * (b * c)) by GROUP_1:def 3
    .= b" * (a" * c") * (a * (c * b * b" * c") * (b * c)) by GROUP_1:def 3
    .= b" * (a" * c") * (a * (c * b * (b" * c")) * (b * c)) by GROUP_1:def 3
    .= b" * (a" * c") * (a * (c * (b * (b" * c"))) * (b * c)) by GROUP_1:def 3
    .= b" * (a" * c") * (a * c * (b * (b" * c")) * (b * c)) by GROUP_1:def 3
    .= b" * (a" * c") * (a * c * ((b * (b" * c")) * (b * c))) by GROUP_1:def 3
    .= b" * (a" * c") * (a * c) * ((b * (b" * c")) * (b * c)) by GROUP_1:def 3
    .= b" * ((a" * c") * (a * c)) * ((b * (b" * c")) * (b * c)) by
GROUP_1:def 3
    .= b" * ((a" * c") * (a * c)) * (b * ((b" * c") * (b * c))) by
GROUP_1:def 3
    .= b" * ((a" * c") * (a * c)) * b * ((b" * c") * (b * c)) by GROUP_1:def 3
    .= [.a,c.] |^ b * ((b" * c") * (b * c)) by Th16
    .= [.a,c.] |^ b * [.b,c.] by Th16;
end;
